Sfera, tekislik va giperbolik tekislikdagi bir tekis qiya ro'yxatlar - Lists of uniform tilings on the sphere, plane, and hyperbolic plane - Wikipedia

Yilda geometriya, sfera, evklid tekisligi va giperbolik tekislikda ko'plab bir xil tekisliklarni bajarish mumkin Wythoff qurilishi tri / p, π / q va π / r kabi ichki burchaklar bilan aniqlangan (p q r) asosiy uchburchak ichida. Maxsus holatlar - bu to'rtburchaklar (p q 2). Yagona echimlar bitta generator punkti tomonidan hayoliy uchburchak ichida 7 ta o'ringa, 3 burchakka, 3 qirraga va uchburchakning ichki qismiga o'rnatiladi. Barcha tepaliklar generatorda yoki uning aks etgan nusxasida mavjud. Chegaralar generator nuqtasi va uning oynadagi tasviri o'rtasida mavjud. Asosiy uchburchakning markazida joylashgan 3 tagacha yuz turi mavjud. To'g'ri uchburchak domenlari 1 ta yuz turiga ega bo'lishi mumkin, ular odatiy shakllarni yaratadilar, umumiy uchburchaklar esa kamida 2 ta uchburchak turlariga ega bo'lib, eng yaxshisi kvazirgular plitkalarga olib keladi.

Ushbu yagona echimlarni ifodalash uchun turli xil belgilar mavjud, Wythoff belgisi, Kokseter diagrammasi, va Kokseterning t-notasi.

Oddiy plitkalar tomonidan ishlab chiqarilgan Mobius uchburchagi p, q, r, butun sonlari bilan Shvarts uchburchagi p, q, r ratsional sonlariga ruxsat bering va ruxsat bering yulduz ko'pburchagi yuzlari va bir-birining ustiga chiqadigan elementlari bor.

7 generator punkti

Har bir to'plam bilan yettita generator ${ displaystyle p, q, r}$ (va bir nechta maxsus shakllar):

Umumiy				To'g'ri uchburchak (r = 2)
Tavsif	Wythoff belgi	Tepalik konfiguratsiya	Kokseter diagramma	Wythoff belgi	Tepalik konfiguratsiya	Schläfli belgi		Kokseter diagramma
muntazam va quasiregular	$q \| p r$	$(p . r) q$		$q \| p 2$	$p q$	{p, q}
	$p \| q r$	$(q . r) p$		$p \| q 2$	$q p$	{q, p}
	$r \| p q$	$(q . p) r$		$2 \| p q$	$(q . p)²$	r {p, q}	t₁{p, q}
kesilgan va kengaytirilgan	$q r \| p$	${ displaystyle q.2p.r.2p}$		$q 2 \| p$	${ displaystyle q.2p.2p}$	t {p, q}	t_0,1{p, q}
	$p r \| q$	${ displaystyle p.2q.r.2q}$		$p 2 \| q$	$p . 2 q .2 q$	t {q, p}	t_0,1{q, p}
	$p q \| r$	${ displaystyle 2r.q.2r.p}$		$p q \| 2$	${ displaystyle 4.q.4.p}$	rr {p, q}	t_0,2{p, q}
tekis yuzli	$p q r \|$	${ displaystyle 2r.2q.2p}$		$p q 2 \|$	${ displaystyle 4.2q.2p}$	tr {p, q}	t_0,1,2{p, q}
tekis yuzli	$p q (r s) \|$	${ displaystyle 2p.2q.-2p.-2q}$	-	$p 2 (r s) \|$	$2 p .4.-2 p . 4 / 3$			-
qotib qolish	$\| p q r$	${ displaystyle 3.r.3.q.3.p}$		$\| p q 2$	${ displaystyle 3.3.q.3.p}$	sr {p, q}
qotib qolish	$\| p q r s$	${ displaystyle (4.p.4.q.4.r.4.s) / 2}$	-	-	-			-

Uchta maxsus holat mavjud:

${ displaystyle p q (r s) |}$ - Bu aralashmasi ${ displaystyle p q r |}$ va ${ displaystyle p q s |}$ , ikkalasi ham birgalikda foydalanadigan yuzlarni o'z ichiga oladi.
${ displaystyle | p q r}$ - Snub shakllari (o'zgaruvchan) ushbu boshqa ishlatilmaydigan belgi bilan beriladi.
${ displaystyle | p q r s}$ - uchun noyob snub shakli U75 bu Wythoff tomonidan tuzilmaydi.

Simmetriya uchburchagi

Ning aks ettirishning 4 ta simmetriya klassi mavjud soha, va uchta Evklid samolyoti. Ulardan bir nechtasi cheksiz ko'p bunday naqshlar giperbolik tekislik shuningdek ro'yxatga olingan. (Giperbolik yoki Evklid plitkalarini belgilaydigan sonlarning birortasini ko'paytirish yana bir giperbolik qoplamani hosil qiladi.)

Nuqta guruhlari:

(p 2 2) dihedral simmetriya, ${ displaystyle p = 2,3,4 nuqta}$ (buyurtma ${ displaystyle 4p}$ )
(3 3 2) tetraedral simmetriya (buyurtma 24)
(4 3 2) oktahedral simmetriya (buyurtma 48)
(5 3 2) ikosahedral simmetriya (buyurtma 120)

Evklid (afin) guruhlari:

(4 4 2) * 442 simmetriya: 45 ° -45 ° -90 ° uchburchak
(6 3 2) *632 simmetriya: 30 ° -60 ° -90 ° uchburchak
(3 3 3) *333 simmetriya: 60 ° -60 ° -60 ° uchburchak

Giperbolik guruhlar:

(7 3 2) *732 simmetriya
(8 3 2) *832 simmetriya
(4 3 3) *433 simmetriya
(4 4 3) *443 simmetriya
(4 4 4) *444 simmetriya
(5 4 2) *542 simmetriya
(6 4 2) *642 simmetriya
...

Ikki tomonlama sferik					Sharsimon
D._{2 soat}	D._{3 soat}	D._{4 soat}	D._{5 soat}	D._{6 soat}	T_d	O_h	Men_h
*222	*322	*422	*522	*622	*332	*432	*532
(2 2 2)	(3 2 2)	(4 2 2)	(5 2 2)	(6 2 2)	(3 3 2)	(4 3 2)	(5 3 2)

Yuqoridagi simmetriya guruhlariga faqat sferadagi butun sonli echimlar kiradi. Shvarts uchburchaklarining ro'yxati ratsional sonlarni o'z ichiga oladi va ularning echimlarining to'liq to'plamini aniqlaydi konveks bo'lmagan bir xil polyhedra.

Evklid samolyoti
p4m	p3m	p6m
*442	*333	*632
(4 4 2)	(3 3 3)	(6 3 2)

Giperbolik tekislik
*732	*542	*433
(7 3 2)	(5 4 2)	(4 3 3)

Yuqoridagi plitkalarda har bir uchburchak juft va g'alati akslantirishlar bilan bo'yalgan asosiy domen hisoblanadi.

Xulosa sharsimon, evklid va giperbolik plitkalar

Wythoff konstruktsiyasi tomonidan yaratilgan tanlangan plitkalar quyida keltirilgan.

Sferik plitkalar (r = 2)

(p q 2)	Ota-ona	Qisqartirilgan	Tuzatilgan	Bitruncated	Birlashtirilgan (dual)	Kantellatsiya qilingan	Hamma narsa (Kantritratsiya qilingan)	Snub
Wythoff belgi	q \| 2-bet	2 q \| p	2 \| p q	2 p \| q	p \| q 2	p q \| 2018-04-02 121 2	p q 2 \|	\| p q 2
Schläfli belgi	${ displaystyle { begin {Bmatrix} p, q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p, q end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} q, p end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} q, p end {Bmatrix}}}$	${ displaystyle r { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle s { begin {Bmatrix} p q end {Bmatrix}}}$
	{p, q}	t {p, q}	r {p, q}	t {q, p}	{q, p}	rr {p, q}	tr {p, q}	sr {p, q}
	t₀{p, q}	t_0,1{p, q}	t₁{p, q}	t_1,2{p, q}	t₂{p, q}	t_0,2{p, q}	t_0,1,2{p, q}	sr {p, q}
Kokseter diagramma
Tepalik shakli	p^q	q.2p.2p	(p.q)²	p. 2q.2q	q^p	p. 4.q.4	4.2p.2q	3.3.p. 3.q
(3 3 2)	{3,3}	(3.6.6)	(3.3a.3.3a)	(3.6.6)	{3,3}	(3a.4.3b.4)	(4.6a.6b)	(3.3.3a.3.3b)
(4 3 2)	{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4a.4)	(4.6.8)	(3.3.3a.3.4)
(5 3 2)	{5,3}	(3.10.10)	(3.5.3.5)	(5.6.6)	{3,5}	(3.4.5.4)	(4.6.10)	(3.3.3a.3.5)

Ba'zi bir-biriga o'xshash sharsimon plitkalar (r = 2)

To'liq ro'yxat, shu jumladan holatlar uchun r ≠ 2, qarang Shvarts uchburchagi bir xil ko'p qirrali ro'yxati.

Plitkalar ko'rsatilgan polyhedra. Ba'zi shakllar buzilib ketgan, ular uchun qavslar bilan berilgan tepalik raqamlari, bir-birining ustiga chiqib ketgan qirralar yoki tepaliklar bilan.

(p q 2)	Ota-ona	Qisqartirilgan	Tuzatilgan	Bitruncated	Birlashtirilgan (dual)	Kantellatsiya qilingan	Hamma narsa (Kantritratsiya qilingan)	Snub
Wythoff belgisi	q \| 2-bet	2 q \| p	2 \| p q	2 p \| q	p \| q 2	p q \| 2018-04-02 121 2	p q 2 \|	\| p q 2
Schläfli belgisi	${ displaystyle { begin {Bmatrix} p, q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p, q end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} q, p end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} q, p end {Bmatrix}}}$	${ displaystyle r { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle s { begin {Bmatrix} p q end {Bmatrix}}}$
	{p, q}	t {p, q}	r {p, q}	t {q, p}	{q, p}	rr {p, q}	tr {p, q}	sr {p, q}
	t₀{p, q}	t_0,1{p, q}	t₁{p, q}	t_1,2{p, q}	t₂{p, q}	t_0,2{p, q}	t_0,1,2{p, q}	sr {p, q}
Kokseter - Dinkin diagrammasi
Tepalik shakli	p^q	(q.2p.2p)	(p.q.p.q)	(2q.2q bet)	q^p	(4.q.4-bet).	(4.2p.2q)	(3.3p. 3.q)
Ikosahedral (5/2 3 2)	{3,5/2}	(5/2.6.6)	(3.5/2)²	[3.10/2.10/2]	{5/2,3}	[3.4.5/2.4]	[4.10/2.6]	(3.3.3.3.5/2)
Ikosahedral (5 5/2 2)	{5,5/2}	(5/2.10.10)	(5/2.5)²	[5.10/2.10/2]	{5/2,5}	(5/2.4.5.4)	[4.10/2.10]	(3.3.5/2.3.5)

Dihedral simmetriya (q = r = 2)

Bilan sferik plitkalar dihedral simmetriya hamma uchun mavjud ${ displaystyle p = 2,3,4, nuqta}$ ko'plari bilan digon degenerativ polyhedraga aylanadigan yuzlar. Sakkiz shakldan ikkitasi (Rektifikatsiya qilingan va kantillangan) replikatsiya bo'lib, jadvalda o'tkazib yuborilgan.

(p 2 2) Asosiy domen	Ota-ona	Qisqartirilgan	Bitruncated (qisqartirilgan dual)	Birlashtirilgan (dual)	Hamma narsa (Kantritratsiya qilingan)	Snub
Kengaytirilgan Schläfli belgisi	${ displaystyle { begin {Bmatrix} p, 2 end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p, 2 end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} 2, p end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} 2, p end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p 2 end {Bmatrix}}}$	${ displaystyle s { begin {Bmatrix} p 2 end {Bmatrix}}}$
	{p, 2}	t {p, 2}	t {2, p}	{2, p}	tr {p, 2}	sr {p, 2}
	t₀{p, 2}	t_0,1{p, 2}	t_1,2{p, 2}	t₂{p, 2}	t_0,1,2{p, 2}	sr {p, 2}
Wythoff belgisi	2 \| 2-bet	2 2 \| p	2 p \| 2018-04-02 121 2	p \| 2018-04-02 2 121 2	p 2 2 \|	\| p 2 2
Kokseter - Dinkin diagrammasi
Tepalik shakli	p²	(2.2p.2p)	(4.4.p)	2^p	(4.2p.4)	(3.3-bet 3)
(2 2 2) V2.2.2	{2,2}	2.4.4	4.4.2	{2,2}	4.4.4	3.3.3.2
(3 2 2) V3.2.2	{3,2}	2.6.6	4.4.3	{2,3}	4.4.6	3.3.3.3
(4 2 2) V4.2.2	{4,2}	2.8.8	4.4.4	{2,4}	4.4.8	3.3.3.4
(5 2 2) V5.2.2	{5,2}	2.10.10	4.4.5	{2,5}	4.4.10	3.3.3.5
(6 2 2) V6.2.2	{6,2}	2.12.12	4.4.6	{2,6}	4.4.12	3.3.3.6
...

Evklid va giperbolik plitkalar (r = 2)

Ba'zi bir giperbolik plitkalar berilgan va a shaklida ko'rsatilgan Poincaré disk proektsiya.

(p q 2)	Jamg'arma. uchburchaklar	Ota-ona	Qisqartirilgan	Tuzatilgan	Bitruncated	Birlashtirilgan (dual)	Kantellatsiya qilingan	Hamma narsa (Kantritratsiya qilingan)	Snub
Wythoff belgisi		q \| 2-bet	2 q \| p	2 \| p q	2 p \| q	p \| q 2	p q \| 2018-04-02 121 2	p q 2 \|	\| p q 2
Schläfli belgisi		${ displaystyle { begin {Bmatrix} p, q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p, q end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} q, p end {Bmatrix}}}$	${ displaystyle { begin {Bmatrix} q, p end {Bmatrix}}}$	${ displaystyle r { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle t { begin {Bmatrix} p q end {Bmatrix}}}$	${ displaystyle s { begin {Bmatrix} p q end {Bmatrix}}}$
		{p, q}	t {p, q}	r {p, q}	t {q, p}	{q, p}	rr {p, q}	tr {p, q}	sr {p, q}
		t₀{p, q}	t_0,1{p, q}	t₁{p, q}	t_1,2{p, q}	t₂{p, q}	t_0,2{p, q}	t_0,1,2{p, q}	sr {p, q}
Kokseter - Dinkin diagrammasi
Tepalik shakli		p^q	(q.2p.2p)	(p.q.p.q)	(2q.2q bet)	q^p	(4.q.4-bet).	(4.2p.2q)	(3.3p. 3.q)
Olti burchakli plitka (6 3 2)	V4.6.12	{6,3}	3.12.12	3.6.3.6	6.6.6	{3,6}	3.4.6.4	4.6.12	3.3.3.3.6
(Giperbolik tekislik) (7 3 2)	V4.6.14	{7,3}	3.14.14	3.7.3.7	7.6.6	{3,7}	3.4.7.4	4.6.14	3.3.3.3.7
(Giperbolik tekislik) (8 3 2)	V4.6.16	{8,3}	3.16.16	3.8.3.8	8.6.6	{3,8}	3.4.8.4	4.6.16	3.3.3.3.8
Kvadrat plitka (4 4 2)	V4.8.8	{4,4}	4.8.8	4.4a.4.4a	4.8.8	{4,4}	4.4a.4b.4a	4.8.8	3.3.4a.3.4b
(Giperbolik tekislik) (5 4 2)	V4.8.10	{5,4}	4.10.10	4.5.4.5	5.8.8	{4,5}	4.4.5.4	4.8.10	3.3.4.3.5
(Giperbolik tekislik) (6 4 2)	V4.8.12	{6,4}	4.12.12	4.6.4.6	6.8.8	{4,6}	4.4.6.4	4.8.12	3.3.4.3.6
(Giperbolik tekislik) (7 4 2)	V4.8.14	{7,4}	4.14.14	4.7.4.7	7.8.8	{4,7}	4.4.7.4	4.8.14	3.3.4.3.7
(Giperbolik tekislik) (8 4 2)	V4.8.16	{8,4}	4.16.16	4.8.4.8	8.8.8	{4,8}	4.4.8.4	4.8.16	3.3.4.3.8
(Giperbolik tekislik) (5 5 2)	V4.10.10	{5,5}	5.10.10	5.5.5.5	5.10.10	{5,5}	5.4.5.4	4.10.10	3.3.5.3.5
(Giperbolik tekislik) (6 5 2)	V4.10.12	{6,5}	5.12.12	5.6.5.6	6.10.10	{5,6}	5.4.6.4	4.10.12	3.3.5.3.6
(Giperbolik tekislik) (7 5 2)	V4.10.14	{7,5}	5.14.14	5.7.5.7	7.10.10	{5,7}	5.4.7.4	4.10.14	3.3.5.3.7
(Giperbolik tekislik) (8 5 2)	V4.10.16	{8,5}	5.16.16	5.8.5.8	8.10.10	{5,8}	5.4.8.4	4.10.16	3.3.5.3.8
(Giperbolik tekislik) (6 6 2)	V4.12.12	{6,6}	6.12.12	6.6.6.6	6.12.12	{6,6}	6.4.6.4	4.12.12	3.3.6.3.6
(Giperbolik tekislik) (7 6 2)	V4.12.14	{7,6}	6.14.14	6.7.6.7	7.12.12	{6,7}	6.4.7.4	4.12.14	3.3.6.3.7
(Giperbolik tekislik) (8 6 2)	V4.12.16	{8,6}	6.16.16	6.8.6.8	8.12.12	{6,8}	6.4.8.4	4.12.16	3.3.6.3.8
(Giperbolik tekislik) (7 7 2)	V4.14.14	{7,7}	7.14.14	7.7.7.7	7.14.14	{7,7}	7.4.7.4	4.14.14	3.3.7.3.7
(Giperbolik tekislik) (8 7 2)	V4.14.16	{8,7}	7.16.16	7.8.7.8	8.14.14	{7,8}	7.4.8.4	4.14.16	3.3.7.3.8
(Giperbolik tekislik) (8 8 2)	V4.16.16	{8,8}	8.16.16	8.8.8.8	8.16.16	{8,8}	8.4.8.4	4.16.16	3.3.8.3.8
(Giperbolik tekislik) (∞ 3 2)	V4.6.∞	{∞,3}	3.∞.∞	3.∞.3.∞	∞.6.6	{3,∞}	3.4.∞.4	4.6.∞	3.3.3.3.∞
(Giperbolik tekislik) (∞ 4 2)	V4.8.∞	{∞,4}	4.∞.∞	4.∞.4.∞	∞.8.8	{4,∞}	4.4.∞.4	4.8.∞	3.3.4.3.∞
(Giperbolik tekislik) (∞ 5 2)	V4.10.∞	{∞,5}	5.∞.∞	5.∞.5.∞	∞.10.10	{5,∞}	5.4.∞.4	4.10.∞	3.3.5.3.∞
(Giperbolik tekislik) (∞ 6 2)	V4.12.∞	{∞,6}	6.∞.∞	6.∞.6.∞	∞.12.12	{6,∞}	6.4.∞.4	4.12.∞	3.3.6.3.∞
(Giperbolik tekislik) (∞ 7 2)	V4.14.∞	{∞,7}	7.∞.∞	7.∞.7.∞	∞.14.14	{7,∞}	7.4.∞.4	4.14.∞	3.3.7.3.∞
(Giperbolik tekislik) (∞ 8 2)	V4.16.∞	{∞,8}	8.∞.∞	8.∞.8.∞	∞.16.16	{8,∞}	8.4.∞.4	4.16.∞	3.3.8.3.∞
(Giperbolik tekislik) (∞ ∞ 2)	V4.∞.∞	{∞,∞}	∞.∞.∞	∞.∞.∞.∞	∞.∞.∞	{∞,∞}	∞.4.∞.4	4.∞.∞	3.3.∞.3.∞

Evklid va giperbolik plitkalar (r > 2)

The Kokseter - Dinkin diagrammasi chiziqli shaklda berilgan, garchi u aslida uchburchak bo'lsa ham, oxirgi segment r birinchi tugunga ulanadi.

Wythoff belgisi (p q r)	Jamg'arma. uchburchaklar	q \| p r	r q \| p	r \| p q	r p \| q	p \| q r	p q \| r	p q r \|	\| p q r
Schläfli belgisi		(p, q, r)	r (r, q, p)	(q, r, p)	r (p, q, r)	(q, p, r)	r (p, r, q)	tr (p, q, r)	s (p, q, r)
Schläfli belgisi		t₀(p, q, r)	t_0,1(p, q, r)	t₁(p, q, r)	t_1,2(p, q, r)	t₂(p, q, r)	t_0,2(p, q, r)	t_0,1,2(p, q, r)	s (p, q, r)
Kokseter diagrammasi
Tepalik shakli		(p.r)^q	(r.2p.q.2p)	(p.q)^r	(q.2r.p. 2r)	(q.r)^p	(2r.q.2r-bet)	(2p.2q.2r)	(3.r.3.q.3.p)
Evklid (3 3 3)	V6.6.6	(3.3)³	3.6.3.6	(3.3)³	3.6.3.6	(3.3)³	3.6.3.6	6.6.6	3.3.3.3.3.3
Giperbolik (4 3 3)	V6.6.8	(3.4)³	3.8.3.8	(3.4)³	3.6.4.6	(3.3)⁴	3.6.4.6	6.6.8	3.3.3.3.3.4
Giperbolik (4 4 3)	V6.8.8	(3.4)⁴	3.8.4.8	(4.4)³	3.8.4.8	(3.4)⁴	4.6.4.6	6.8.8	3.3.3.4.3.4
Giperbolik (4 4 4)	V8.8.8	(4.4)⁴	4.8.4.8	(4.4)⁴	4.8.4.8	(4.4)⁴	4.8.4.8	8.8.8	3.4.3.4.3.4
Giperbolik (5 3 3)	V6.6.10	(3.5)³	3.10.3.10	(3.5)³	3.6.5.6	(3.3)⁵	3.6.5.6	6.6.10	3.3.3.3.3.5
Giperbolik (5 4 3)	V6.8.10	(3.5)⁴	3.10.4.10	(4.5)³	3.8.5.8	(3.4)⁵	4.6.5.6	6.8.10	3.5.3.4.3.3
Giperbolik (5 4 4)	V8.8.10	(4.5)⁴	4.10.4.10	(4.5)⁴	4.8.5.8	(4.4)⁵	4.8.5.8	8.8.10	3.4.3.4.3.5
Giperbolik (6 3 3)	V6.6.12	(3.6)³	3.12.3.12	(3.6)³	3.6.6.6	(3.3)⁶	3.6.6.6	6.6.12	3.3.3.3.3.6
Giperbolik (6 4 3)	V6.8.12	(3.6)⁴	3.12.4.12	(4.6)³	3.8.6.8	(3.4)⁶	4.6.6.6	6.8.12	3.6.3.4.3.3
Giperbolik (6 4 4)	V8.8.12	(4.6)⁴	4.12.4.12	(4.6)⁴	4.8.6.8	(4.4)⁶	4.8.6.8	8.8.12	3.6.3.4.3.4
Giperbolik (∞ 3 3)	V6.6.∞	(3.∞)³	3.∞.3.∞	(3.∞)³	3.6.∞.6	(3.3)^∞	3.6.∞.6	6.6.∞	3.3.3.3.3.∞
Giperbolik (∞ 4 3)	V6.8.∞	(3.∞)⁴	3.∞.4.∞	(4.∞)³	3.8.∞.8	(3.4)^∞	4.6.∞.6	6.8.∞	3.∞.3.4.3.3
Giperbolik (∞ 4 4)	V8.8.∞	(4.∞)⁴	4.∞.4.∞	(4.∞)⁴	4.8.∞.8	(4.4)^∞	4.8.∞.8	8.8.∞	3.∞.3.4.3.4
Giperbolik (∞ ∞ 3)	V6.∞.∞	(3.∞)^∞	3.∞.∞.∞	(∞.∞)³	3.∞.∞.∞	(3.∞)^∞	∞.6.∞.6	6.∞.∞	3.∞.3.∞.3.3
Giperbolik (∞ ∞ 4)	V8.∞.∞	(4.∞)^∞	4.∞.∞.∞	(∞.∞)⁴	4.∞.∞.∞	(4.∞)^∞	∞.8.∞.8	8.∞.∞	3.∞.3.∞.3.4
Giperbolik (∞ ∞ ∞)	V∞.∞.∞	(∞.∞)^∞	∞.∞.∞.∞	(∞.∞)^∞	∞.∞.∞.∞	(∞.∞)^∞	∞.∞.∞.∞	∞.∞.∞	3.∞.3.∞.3.∞

Shuningdek qarang

Adabiyotlar

Kokseter Muntazam Polytopes, Uchinchi nashr, (1973), Dover nashri, ISBN 0-486-61480-8 (V bob: Kaleydoskop, Bo'lim: 5.7 Vaytof qurilishi)
Kokseter Geometriyaning go'zalligi: o'n ikkita esse, Dover Publications, 1999, ISBN 0-486-40919-8 (3-bob: Uythoffning yagona politoplar uchun qurilishi)
Kokseter, Longuet-Xiggins, Miller, Yagona polyhedra, Fil. Trans. 1954, 246 A, 401-50.
Venninger, Magnus (1974). Polyhedron modellari. Kembrij universiteti matbuoti. ISBN 0-521-09859-9. 9-10 betlar.

Tashqi havolalar

Vayshteyn, Erik V. "Wythoff belgisi". MathWorld.
Wythoff belgisi
Wythoff belgisi^{[doimiy o'lik havola ]}
Wythoff qurilish uslubidan foydalangan holda bir xil polidralarni namoyish qilish uchun Greg Eganning appleti
Vaythoffning qurilish usulini Shadertoy taqdim etadi
KaleidoTile 3 Tomonidan Windows uchun bepul o'quv dasturi Jeffri Uiks sahifadagi ko'plab rasmlarni yaratgan.
Xetch, Don. "Giperbolik planar tessellations".

Ushbu maqola matematikaga oid narsalarni o'z ichiga oladi ro'yxatlar ro'yxati.